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1 | (30) |
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1.1 Some Basic Mathematical Models; Direction Fields |
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1 | (9) |
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1.2 Solutions of Some Differential Equations |
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10 | (9) |
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1.3 Classification of Differential Equations |
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19 | (7) |
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26 | (5) |
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Chapter 2 First Order Differential Equations |
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31 | (106) |
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2.1 Linear Equations; Method of Integrating Factors |
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31 | (11) |
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42 | (9) |
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2.3 Modeling with First Order Equations |
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51 | (17) |
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2.4 Differences Between Linear and Nonlinear Equations |
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68 | (10) |
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2.5 Autonomous Equations and Population Dynamics |
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78 | (17) |
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2.6 Exact Equations and Integrating Factors |
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95 | (7) |
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2.7 Numerical Approximations: Euler's Method |
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102 | (10) |
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2.8 The Existence and Uniqueness Theorem |
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112 | (10) |
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2.9 First Order Difference Equations |
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122 | (15) |
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Chapter 3 Second Order Linear Equations |
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137 | (84) |
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3.1 Homogeneous Equations with Constant Coefficients |
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137 | (8) |
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3.2 Solutions of Linear Homogeneous Equations; the Wronskian |
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145 | (13) |
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3.3 Complex Roots of the Characteristic Equation |
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158 | (9) |
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3.4 Repeated Roots; Reduction of Order |
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167 | (8) |
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3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients |
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175 | (11) |
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3.6 Variation of Parameters |
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186 | (6) |
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3.7 Mechanical and Electrical Vibrations |
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192 | (15) |
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207 | (14) |
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Chapter 4 Higher Order Linear Equations |
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221 | (26) |
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4.1 General Theory of nth Order Linear Equations |
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221 | (7) |
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4.2 Homogeneous Equations with Constant Coefficients |
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228 | (8) |
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4.3 The Method of Undetermined Coefficients |
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236 | (5) |
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4.4 The Method of Variation of Parameters |
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241 | (6) |
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Chapter 5 Series Solutions of Second Order Linear Equations |
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247 | (62) |
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5.1 Review of Power Series |
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247 | (7) |
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5.2 Series Solutions Near an Ordinary Point, Part I |
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254 | (11) |
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5.3 Series Solutions Near an Ordinary Point, Part II |
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265 | (7) |
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5.4 Euler Equations; Regular Singular Points |
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272 | (10) |
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5.5 Series Solutions Near a Regular Singular Point, Part I |
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282 | (6) |
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5.6 Series Solutions Near a Regular Singular Point, Part II |
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288 | (8) |
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296 | (13) |
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Chapter 6 The Laplace Transform |
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309 | (50) |
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6.1 Definition of the Laplace Transform |
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309 | (8) |
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6.2 Solution of Initial Value Problems |
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317 | (10) |
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327 | (9) |
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6.4 Differential Equations with Discontinuous Forcing Functions |
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336 | (7) |
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343 | (7) |
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6.6 The Convolution Integral |
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350 | (9) |
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Chapter 7 Systems of First Order Linear Equations |
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359 | (92) |
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359 | (9) |
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368 | (10) |
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7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors |
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378 | (12) |
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7.4 Basic Theory of Systems of First Order Linear Equations |
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390 | (6) |
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7.5 Homogeneous Linear Systems with Constant Coefficients |
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396 | (12) |
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408 | (13) |
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421 | (8) |
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429 | (11) |
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7.9 Nonhomogeneous Linear Systems |
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440 | (11) |
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Chapter 8 Numerical Methods |
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451 | (44) |
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8.1 The Euler or Tangent Line Method |
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451 | (11) |
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8.2 Improvements on the Euler Method |
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462 | (6) |
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8.3 The Runge--Kutta Method |
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468 | (4) |
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472 | (6) |
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8.5 Systems of First Order Equations |
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478 | (4) |
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8.6 More on Errors; Stability |
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482 | (13) |
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Chapter 9 Nonlinear Differential Equations and Stability |
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495 | (94) |
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9.1 The Phase Plane: Linear Systems |
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495 | (13) |
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9.2 Autonomous Systems and Stability |
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508 | (11) |
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9.3 Locally Linear Systems |
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519 | (12) |
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531 | (13) |
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9.5 Predator-Prey Equations |
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544 | (10) |
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9.6 Liapunov's Second Method |
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554 | (11) |
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9.7 Periodic Solutions and Limit Cycles |
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565 | (12) |
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9.8 Chaos and Strange Attractors: The Lorenz Equations |
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577 | (12) |
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Chapter 10 Partial Differential Equations and Fourier Series |
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589 | (88) |
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10.1 Two-Point Boundary Value Problems |
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589 | (7) |
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596 | (11) |
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10.3 The Fourier Convergence Theorem |
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607 | (7) |
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10.4 Even and Odd Functions |
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614 | (9) |
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10.5 Separation of Variables; Heat Conduction in a Rod |
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623 | (9) |
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10.6 Other Heat Conduction Problems |
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632 | (11) |
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10.7 The Wave Equation: Vibrations of an Elastic String |
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643 | (15) |
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658 | (19) |
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Appendix A Derivation of the Heat Conduction Equation |
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669 | (4) |
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Appendix B Derivation of the Wave Equation |
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673 | (4) |
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Chapter 11 Boundary Value Problems and Sturm-Liouville Theory |
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677 | (62) |
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11.1 The Occurrence of Two-Point Boundary Value Problems |
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677 | (8) |
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11.2 Sturm-Liouville Boundary Value Problems |
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685 | (14) |
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11.3 Nonhomogeneous Boundary Value Problems |
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699 | (15) |
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11.4 Singular Sturm-Liouville Problems |
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714 | (7) |
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11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion |
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721 | (7) |
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11.6 Series of Orthogonal Functions: Mean Convergence |
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728 | (11) |
Answers to Problems |
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739 | (60) |
Index |
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799 | |